Journal of Noncommutative Geometry

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Volume 11, Issue 2, 2017, pp. 741–756
DOI: 10.4171/JNCG/11-2-9

Published online: 2017-06-29

Deformation quantization of integrable systems

Georgy Sharygin[1] and Dmitry Talalaev[2]

(1) Lomonosov Moscow State University and ITEP, Moscow, Russia
(2) Lomonosov Moscow State University and ITEP, Moscow, Russia

In this paper we address the following question: is it always possible to choose a deformation quantization of a Poisson algebra $\mathcal{A}$ so that certain Poisson-commutative subalgebra $\mathcal{C}$ in it remains commutative? We define a series of cohomological obstructions to this, that take values in the Hochschild cohomology of $\mathcal{C}$ with coefficients in $\mathcal{A}$. In some particular case of the pair $(\mathcal{A},\mathcal{C})$ we reduce these classes to the classes of the Poisson relative cohomology of the Hochschild cohomology. We show, that in the case, when the algebra $\mathcal{C}$ is polynomial, these obstructions coincide with the previously known ones, those which were defined by Garay and van Straten.

Keywords: Quantization, integrable systems, Hochschild relative cohomology

Sharygin Georgy, Talalaev Dmitry: Deformation quantization of integrable systems. J. Noncommut. Geom. 11 (2017), 741-756. doi: 10.4171/JNCG/11-2-9